Problem: $\dfrac{ -i - 8j }{ 7 } = \dfrac{ i + 4k }{ 9 }$ Solve for $i$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -i - 8j }{ {7} } = \dfrac{ i + 4k }{ 9 }$ ${7} \cdot \dfrac{ -i - 8j }{ {7} } = {7} \cdot \dfrac{ i + 4k }{ 9 }$ $-i - 8j = {7} \cdot \dfrac { i + 4k }{ 9 }$ Multiply both sides by the right denominator. $-i - 8j = 7 \cdot \dfrac{ i + 4k }{ {9} }$ ${9} \cdot \left( -i - 8j \right) = {9} \cdot 7 \cdot \dfrac{ i + 4k }{ {9} }$ ${9} \cdot \left( -i - 8j \right) = 7 \cdot \left( i + 4k \right)$ Distribute both sides ${9} \cdot \left( -i - 8j \right) = {7} \cdot \left( i + 4k \right)$ $-{9}i - {72}j = {7}i + {28}k$ Combine $i$ terms on the left. $-{9i} - 72j = {7i} + 28k$ $-{16i} - 72j = 28k$ Move the $j$ term to the right. $-16i - {72j} = 28k$ $-16i = 28k + {72j}$ Isolate $i$ by dividing both sides by its coefficient. $-{16}i = 28k + 72j$ $i = \dfrac{ 28k + 72j }{ -{16} }$ All of these terms are divisible by $4$ Divide by the common factor and swap signs so the denominator isn't negative. $i = \dfrac{ -{7}k - {18}j }{ {4} }$